August 12, 1897] 



NATURE 



339 



abstract question only when all ideas involved have been 

 ■called into life in the student, is altogether to be praised. 

 Thus a sure foundation is laid. 



There follows next the example of a train " going at 

 thirty miles an houn" The notion of velocity (why not 

 ■call it speed ?), and with it that of acceleration, is made 

 mathematically exact. From the law of falling bodies 

 s = i6t/-, the velocity is next deduced. Here the 

 ■" limiting value " is introduced, and the ordinary process 

 •criticised. To quote Prof. Perry :— " Some people have 

 the notion that we are stating something that is only 

 approximately true ; it is often because their teacher will 

 say such things as 'reject i6"i8/ because it is small,' or 

 *let dt be an infinitely small amount of time,' and they 

 proceed to divide something by it, showing that although 

 they may reach the age of Methuselah they will never 

 have the common sense of an engineer." It is the art 

 of taking for granted what the " common sense of an 

 engineer" — that is, of any man who has been obliged to 

 think seriously about the things before him, and not only 

 about how to fill so many sheets of paper in a certain 

 time with answers — instinctively knows to be true, and of 

 leaving out of account all the considerations which super- 

 fine criticism of minute and exact abstract considerations 

 introduce, which Prof. Perry's book specially emphasises. 



After such preliminaries, interspersed with remarks on 

 many things, such as force and weight, we have the 

 equation y = ax'- in the abstract, and from it dyldx and 

 d^yldx'^ with their integrals and applications to uniformly 

 accelerated motion, the energy of elongated springs, to 

 Ohm's law and to transformers, till at last the case 



is considered. Here the binomial theorem is supposed 

 to be known in its general form. This assumption is 

 one of the few points where a greater simplification 

 might possibly be introduced by giving a few examples 

 where « is 3, ^, - i or - 2, to show that the formulas are 

 right. I believe that the learner would have greater 

 faith in the result obtained by seeing it verified in special 

 cases, and this is easy. In the integration of .1" the case 

 n = - I is here assumed to give log .r, the proof being 

 left to Chapter ii. 



At the end of Chapter i. (nearly half the book) there 

 are applications in the most varied form of the simple 

 function .r" ; and ".r"" forms an appropriate heading to 

 the chapter. Partial differentiation is also introduced 

 as a thing only to be mentioned in order to be under- 

 stood. We are so much accustomed to have most of the 

 simple functions at our disposal from the beginning, that 

 we do not altogether realise how much can be done with 

 the x" alone. The applications given by Prof Perry are 

 of a most varied kmd, very much more so than those in 

 the ordinary text-books, which almost exclusively treat 

 of problems either purely mathematical or geometrical. 

 The first example relates to a perfect steam engine, 

 then come a study of curves, maxima and minima, 

 strength of rectangular beams, electrical problems, areas, 

 volumes, centres of gravity, moments of inertia, curva- 

 ture, bending, fluid motion and level surfaces, magnetic 

 field, the two elasticities, laws of thermodynamics and 

 entropy, only to mention some of the headings which 

 strike the eye in turning over the pages. Differential 

 NO. 1450, VOL. 56] 



equations, ordinary or other, are introduced without 

 hesitation. 



It will be seen from this that the course pursued in 

 this book is very different indeed from that of ordinary 

 text-books. The aim is everywhere to go on slowly with 

 the purely mathematical work, but to make the student 

 feel at each step that he has gained actual and useful 

 knowledge which leads at once to important practical 

 applications ; and also to show from the beginning the 

 naturalness of the processes, and to disabuse the beginner 

 of any preconceived idea that the calculus is brimful of 

 difficult and superfine abstractions. 



The second chapter contains the compound interest 

 law and the harmonic function; it is headed "^ and 

 sin x." The exponential theorem is assumed to be 

 known. This is followed by a new series of applications, 

 including Newton's law of cooling, slipping of a belt over 

 a pulley, and so on ; and some problems are studied of 

 a body vibrating, introducing both forced vibrations and 

 damping. Mathematical formulation of the problem 

 leads to a linear differential equation of the second order 

 with constant coefficients. The problem of two electric 

 currents with resistance and self- and mutual-induction 

 leads to an equation exactly similar ; a solution of one 

 of these problems contains, therefore, that of the other. 

 The mechanical problem, as being more easily followed, 

 is worked out fully ; transformers are also dealt with. 

 In fact, both the mechanical and the electrical engineer 

 will get from the study of these first two chapters a great 

 deal of information on the subject he is interested in. 



There is a third chapter where more compound 

 functions are considered. It will be remembered that in 

 the first two chapters functions of functions are practi- 

 cally left out, and only the simplest and fundamental 

 functions x'\ e" and sin x are introduced. The author 

 now recommends the student to supplement the know- 

 ledge so far gained by reading the ordinary treatises on 

 the calculus ; but he gives a short outline of the results, in 

 his third and last chapter, which has the characteristic 

 heading "Academic Exercises." But even here, in a 

 little more than 100 pages, he goes beyond what is found 

 in most elementary treatises ; for he introduces a good 

 many differential equations, and even zonal harmonics 

 and Bessel functions are touched upon. In the middle 

 of the chapter there is a page devoted to Osborne 

 Reynolds' "Theory of Lubrication of Journals," where 

 the essence of Reynolds' complicated investigation is 

 given in a marvellously simple manner. 



Of course, from a purely mathematical and academic 

 point of view, it would be easy to find fault, and perhaps 

 to condemn, the whole work. But the book is not meant 

 to be academic ; in fact, it is from beginning to end a 

 protest against the academic treatment of mathematics, 

 and as such we welcome it most heartily. 



We recommend the book not only to the engineering 

 student and to all who want to learn the calculus, but, 

 indeed, especially to teachers of the subject, who will find 

 many points raised in it to set them thinking, quite apart 

 from the great variety of examples given. We also re- 

 commend it most strongly to teachers in the modern or 

 science side of secondary schools, h. careful study of 

 the Introduction and the beginning of Chapter i. will 

 . give- them many hints as to how to make mathematics 



